Flows of Conformally Coclosed G 2 -Structures with Dilaton.

Abstract

We study flows of $G_2$ -structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$ -geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$ -Laplacian coflow, which lifts the Kähler-Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$ -lift of the anomaly flow deforms conformally coclosed $G_2$ -structures. We compare the $G_2$ -anomaly flow to the $G_2$ -Laplacian coflow, and investigate short-time existence and fixed points.